MAT3633

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Course Catalog

MAT 3633. Numerical Analysis. (3-0) 3 Credit Hours.

Prerequisites: MAT2233, MAT3213, and one of the following: CS1063, CS1714, or CS2073. Solution of linear and nonlinear equations, curve-fitting, and eigenvalue problems. Generally offered: Fall, Spring. Differential Tuition: $150.

Topics List

Date Sections Topics Prerequisite Skills Student Learning Outcomes
Week 1

Section 0.2: Loss of significant digits

  • Nested Multiplication for Evaluating Polynomials
  • Machine Representation of Real Numbers
  • Loss of Significant Digits in Numerical Computing
  • Review of Taylor's Theorem
Week 1

Section 1.1: Fixed-Point Iteration

  • Bisection Method and Implementation
  • Brief Introduction to Matlab
Week 2

Section 1.2: Fixed-Point Iteration

  • Geometric Interpretation of Fixed-Point Iteration
  • Convergence of Fixed Point Iterations
  • Order of Convergence of Iterative Methods
Week 2

Section 1.3: Limits of Accuracy: Conditioning of Problems

  • Sensitivity Analysis of Root-Finding
  • Error Magnification Factor for Solution of Equations
Week 3

Section 1.4: Newton's Method

  • Algebraic and Geometric Interpretation of Newton's method
  • Error Analysis for Newton's Method Based on Taylor's Theorem
  • Newton's Method as a Fixed Point Iteration
  • Modified Newton's Method and its Rate of Convergence
Week 3

Section 1.5 Root-Finding Without Derivatives

  • Secant Method and its Convergence
  • Stopping Criteria for Iterative Methods
Week 4

Section 2.1 Solve Systems of Linear Equations: Gaussian Elimination

  • Gaussian Elimination and its Operation Counts
  • Gaussian Elimination with Pivoting
  • Implementation of Gauss Elimination
Week 4

Section 2.2 Solve Systems of Linear Equations: LU Decomposition

  • Matrices for Elementary Row Operations
  • Gauss Elimination as Matrix Products
  • Advantages of Solutions by LU Decomposition
Week 5

Section 2.3 Error Analysis for Solution of Ax=b

  • Various Norms for Vectors and Matrices: Compatibility of Vector and Matrix Norms
  • Error Analysis for Solution of Ax=b
  • Error Magnification Factor and Condition Number of Matrix
Week 5

Section 2.5: Iterative Methods for Solving Ax=b

  • Convergence of General Iterative Method for Solving System of Linear Equations
  • Comparison of Gauss Elimination and Iterative Methods
Week 6

Section 2.6: Conjugate Gradient (CG) Method

  • Symmetric Positive Definite Matrix and Properties
  • Construction of Conjugate Gradient (CG) Method
  • Properties of CG Method
  • Preconditioning for CG Method
Week 6

Section 2.7: Nonlinear System of Equations

  • (TBD)
Week 7

Sections 3.1: Data and Interpolating Functions

  • Properties of Lagrange Basis Functions
  • Lagrange Form of the Interpolation Polynomials
  • Properties of Newton's Divided Differences
  • Newton's Form of the Interpolation Polynomials
Week 7

Section 3.2: Interpolation Error and Runge Phenomenon

  • (TBD)
Week 8

Section 3.4: Cubic Splines

  • Construction of Cubic Splines for Interpolation
  • End Conditions
  • Properties of Cubic Spline Interpolation
Week 8

Section 3.5: Bezier Curves

  • Bezier Curve and Fonts
Week 8

Section 4.1: Least Square Method

  • Least Square Method for Solving Inconsistent System of Linear Equations]
  • Basic Properties of Least Square Solutions
Week 9

Section 4.2: Mathematical Models and Data Fitting

  • Least square method for curve fitting and statistical modeling
  • Survey of Models: linear model, periodic model, exponential models, logistic model, etc
Week 9

Section 4.5: Nonlinear Least Square Fitting

  • (TBD)
Week 10

Section 5.1: Numerical Differentiation

  • Finite Difference (FD) Approximations of 1st order Derivative and Their Error Analysis
  • FD approximations of 2nd order Derivatives and Their Error Analysis
  • Undetermined Coefficient Method for FD Approximation
  • Extrapolation Technique for Improving the Order of Approximation
Week 10

Section 5.2: Numerical Integration: Newton-Cotes Formulas

  • Error Analysis based on Taylor's Theorem
  • Error Analysis based on Interpolation Errors
  • Degree of Precision of Quadrature Rules
Week 10

Section 5.3: Numerical Integration: Romberg's Technique

  • Motivation, construction and implementation of Romberg's Technique.
Week 11

Section 5.4: Adaptive Numerical Integration

  • How to estimate the error on a sub interval
  • How to mark sub intervals to be further refinement?
Week 11

Section 5.5: Gauss Quadrature Formulas

  • Motivation and difficulties with straightforward approach
  • Legendre polynomials and their basic properties
  • Gauss Quadrature rule based on Legendre polynomials
  • Degree of precision of Gauss Quadrature
  • Gauss quadrature formula on general interval and composite Gauss rules
Week 12

Section 10.1: Discrete Fourier Transform and Fast Fourier Transform (FTT)

  • Matrix Form of Discrete Fourier Transform
  • DFT and Trigonometric Interpolation
Week 12

Section 11.1: Discrete Cosine Transform (optional)

  • DCT and Interpolation by Cosine Functions
  • Relation between DFT and DCT
  • Fourier Transform of 2-Dimensional Functions
  • DCT of 2-Dimensional Functions
  • Interpolation Theorem for 2-Dimensional DCT
Week 12

Section 11.2: Image Compression (optional)

  • Digital Gray scale images and color color images
  • RGB format
  • YCbCr (or YUV) format
  • Convertion between RGB and YUV formats
Week 13

Section 12.1: Power Iteration Methods

  • Convergence of Power Iteration Methods
Week 13

Section 12.2: QR Algorithm for Computing Eigenvalues

  • Definition and basic properties of orthogonal matrices
  • QR-Factorization based on Gram-Schmidt Orthogonalization
Week 14

Section 12.2: QR Algorithm for Computing Eigenvalues

  • Convert a matrix into UHF by Householder reflectors